Boundedness in a parabolic–elliptic chemotaxis-growth system under a critical parameter condition

We consider the parabolic–elliptic chemotaxis-growth system {ut=Δu−χ∇⋅(um∇v)+μu(1−uα),x∈Ω,t>0,−Δv+v=uγ,x∈Ω,t>0, under no-flux boundary conditions in a smoothly bounded domain Ω⊂RN, N≥1N≥1, where χ,μ,m,αχ,μ,m,α and γγ are prescribed positive parameters fulfilling m≥1m≥1 and γ≥1γ≥1.Recently, it has been proved in Galakhov et al. (2016) that if either α>m+γ−1α>m+γ−1 or α=m+γ−1α=m+γ−1 and μ>Nα−22(m−1)+Nαχ, for any given u0∈W1,∞(Ω) this system possesses a global and bounded classical solution. The present work further shows that the same conclusion still holds for the critical case α=m+γ−1α=m+γ−1 and μ=Nα−22(m−1)+Nαχ.